Basic Concepts in Nonlocal Approaches between Vr, a, and a

K(x) –= / a-(jx – s|)c/l/(s) and d(x, s) ~ (13.1.2)

Jv Vr (X)

The function a(r) decays with the distance from point x and is zero or nearly zero at points sufficiently remote from x. The simplest is the uniform averaging function, for which a = 1 in a sphere of diameter l and 0 outside. For points in the interior of the body whose distance to any boundary is larger than I/2, Vr is the volume of the averaging sphere Vr — trt3/6. However, for points closer to the surface, the part of the sphere that protrudes outside the body does not contribute to the integral in (13.1.2) and Vr must be considered a variable (Fig. 13.1.1c).

We may note that according to (13.1.1) and (13.1.2), a may be multiplied by an arbitrary factor without introducing any change in the nonlocal variable, because Vr also gets multiplied by the same factor. This means that we can rescale a at will. In the following, we scale « so that the value of a at the origin is 1, i. e„

a(0) – I (13.1.3)

as depicted in Fig. 13.1. Id.

The convergence of numerical solutions is slightly better if a is a smooth bell-shaped function (Fig. 13.1.Id, full line) rather than rectangular (Fig. 13.1.1 d, dashed line). According to Bazant (1990c), an effective choice is the function

Of = [l–(r/p</)2]" if r<pot, a ~ 0 if |r| > pot (13.1.4)

where r — |x — s| is the distance from point x, £ is the characteristic length (a material property, Fig. 13.1.1), and pa is a coefficient chosen in such a manner that the volume under function a given by Eq. (13.1.4) is equal to the volume of the uniform distribution in Fig. 13.1.Id. Front this requirement, one may calculate that pa — v^35/4 — 0.8178. In the earlier works, the normal (Gaussian) distribution function has also been used instead of Eq. (13.1.4) and was found to work well, although its values are nowhere exactly zero. Note that the limit of nonlocal continuum for^ —> 0 is the local continuum (because £->£)•

The foregoing approximation deals with three-dimensional averaging. In many cases, however, two – or one-dimensional approximations are required. In those cases, similar definitions can be written for the averaging operator. For the two-dimensional case Vr must be replaced by Ar, a representative area, and the integrals become surface integrals. For the uniaxial case the integrals reduce to simple integrals and Vr is replaced by a reference length LT. It is an easy matter to see that if the size of the representative zone is t in either dimension, and the averaging is uniform, then, for interior points Ar — tt£2/4 and Lr — l. For the bell-shaped function (13.1.4), the values of pa to be used for two and one dimensions are adjusted so that they give the same values for AT and as the uniform distribution (see the exercises at the end of this section).

Now that we have introduced the concept of nonlocal averaging, formulating the equations of a nonlocal continuum seems to be a simple matter: Just replace some or all of the classical local variables by their nonlocal averages. However, this is not easy because, in general, some physically problematic features appear and the model does not work at all. In the remainder of this section we focus on uniaxial models to illustrate some of the problems that may arise and the approaches devised to overcome them, leading to various useful models.